Problem 6 Test the hypothesis, using (a) t... [FREE SOLUTION]

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Chapter 10: Problem 6

Test the hypothesis, using (a) the classical approach and then (b) the P-value approach. Be sure to verify the requirements of the test. $$\begin{aligned}&H_{0}: p=0.25 \text { versus } H_{1}: p<0.25\\\&n=400 ; x=96 ; \alpha=0.1\end{aligned}$$

Short Answer

Expert verified

Fail to reject $ H_0 $ with test statistic $ z = -0.5774 $ and P-value = 0.2810.

Step by step solution

- State the hypotheses

The null hypothesis ($H_0$): $p = 0.25$. The alternative hypothesis ($H_1$) is $p < 0.25$.

- Check the requirements

Ensure the sample size is large enough and that the sampling distribution of $\bar{p}$ follows a normal distribution. $n = 400$ and $np = 400 \times 0.25 = 100 \text{ and } n(1-p) = 400 \times 0.75 = 300$. Both values are > 5, so the normal approximation can be used.

- Calculate the test statistic (Classical approach)

Calculate the sample proportion $\bar{p} = \frac{x}{n} = \frac{96}{400} = 0.24$. Next, find the test statistic using $z = \frac{\bar{p} - p}{\frac{\text{value}\text{ } \text{of}\text{ }p\text{ } \bar{p}(1-p) / n}} = \frac{0.24 - 0.25}{\frac{0.25 \times 0.75}{400}} = -0.5774$

- Determine the critical value

Considering $ \text{ standard normal table for } \text{ alpha } = 0.1 \text{, the critical value } z_{0.1} = -1.28$.

- Make a decision (Classical approach)

Since the test statistic $z = -0.5774$ is greater than the critical value $z_{0.1} = -1.28$, we fail to reject $ H_0$.

- Calculate the P-value (P-value approach)

Find the P-value corresponding to $z = -0.5774$ from the standard normal distribution table. $ P( Z < -0.5774 ) = 0.2810$. This is higher than $ \text{ } \text{ alpha } = 0.1$, so $ H_0$ is not rejected.

- Conclude based on P-value approach

Since the P-value $ 0.2810 \text{ is greater than alpha } \text{ } 0.1$, we fail to reject the null hypothesis $ H_0$ using the P-value approach as well.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis, often denoted as $H_0$, is a statement that there is no effect or difference. In hypothesis testing, it's the assumption that any observed effect is due to chance. For this exercise, the null hypothesis is $p = 0.25$. This means that we assume the population proportion is 0.25 unless proven otherwise by our test.

Alternative Hypothesis

The alternative hypothesis, denoted as $H_1$, is the statement we want to test against the null hypothesis. It represents an effect or difference. Here, the alternative hypothesis is $p < 0.25$, suggesting that the population proportion is less than 0.25. Unlike the null, this needs to be determined by evidence from our sample data.

Normal Distribution

Normal distribution is a continuous probability distribution that plays a key role in hypothesis testing. It's often called the bell curve due to its shape. For our test to be valid, the sampling distribution of the sample proportion $\bar{p}$ must follow a normal distribution.
To check this, we ensure that both $np$ and $n(1-p)$ are greater than 5. In our example:

$n \cdot p = 400 \cdot 0.25 = 100$
$n \cdot (1-p) = 400 \cdot 0.75 = 300$

Both values are indeed greater than 5, so we can use the normal distribution for this hypothesis test.

Test Statistic

A test statistic is a standardized value that we calculate from sample data during a hypothesis test. It helps us decide whether to reject the null hypothesis. For this exercise, we calculate the test statistic using the formula for a proportion:

Calculate the sample proportion: $ \bar{p} = \frac{x}{n} = \frac{96}{400} = 0.24 $
Compute the standard error for $ \bar{p}$: $\text{SE} = \frac{\text{value of} p \times (1-p)}{n} = \frac{0.25 \times 0.75}{400}$
Find the test statistic: $ z = \frac{ \bar{p} - p }{ \text{SE} } = \frac{0.24 - 0.25}{\frac{0.25 \times 0.75}{400}} = -0.5774 $

P-value

The P-value is the probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is true. It's a measure that helps us understand whether to reject the null hypothesis. To find the P-value for our test statistic:

Look up the test statistic $z = -0.5774$ in the standard normal distribution table.
The P-value for $ z = -0.5774 $ is approximately 0.2810.

Since our P-value (0.2810) is greater than the significance level $ \alpha = 0.1 $, we fail to reject the null hypothesis. This means there isn't sufficient evidence to support the alternative hypothesis that $ p < 0.25 $.

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